In principal-agent models, collusion among agents generally lowers
the principal's welfare in the presence of asymmetric information.
Under collusion, the agents can have more opportunities to take
advantage of information possessed only by them and not by the
principal. Standard treatment for collusion in the literature is to
"deter it" if affordable and to "allow it"
otherwise. Although a few studies show that the prospect of collusion
can be beneficial for the principal ex ante, a common result in hidden
information models is that when side contracting among agents takes
place, the agents' information is not revealed to the principal. In
this paper, we examine a situation in which the principal may take
advantage of collusion among agents when inducing revelation of hidden
information.

We consider an organization in which the top management has limited
power to discriminate transfers to different subunits, while it observes
each subunit's performance perfectly. An organization facing such a
restriction is said to be under "external wage compression."
(1) Our results suggest that, under wage compression, the top management
may increase efficiency in the output schedule by inducing collusion
among subunits. For simplicity, we model the situation with one
principal and two agents. Each agent's "type," such as
his efficiency or cost parameter, is not known to the principal, and
prior to production, each agent reports his type to the principal. An
agent can quit if he anticipates a payoff strictly less than his
reservation payoff. In our model, externalities due to wage compression
provide a potential free ride when the types of the agents are
different--the less efficient agent can free ride on the more efficient
agent's production.

This free-riding opportunity due to limited wage discrimination
leads to an incentive problem associated with hidden information. An
efficient agent facing another efficient agent has an incentive to
misreport his type to free ride on the other agent. Thus, when both
agents are the efficient type, they have information rent for the
potential situation in which one of them misreports. When the agents
cannot collude, the principal distorts the allocation of production when
the types reported by the agents are different--in the optimal contract,
the principal increases (decreases) the proportion of the output
assigned to the inefficient (efficient) agent. By doing so, the
principal removes an inefficient agent's free riding on an
efficient agent, which in turn discourages an efficient agent from
misreporting his type when paired with another efficient agent.

Under collusion, however, the principal can improve her payoff for
two reasons. First, she can make agents of different types internalize the externality by inducing side transfers (2) between them. When the
agents are of different types, the inefficient agent needs the efficient
agent to be truthful in order to free ride on him. Therefore, the
inefficient agent has an incentive to offer the efficient agent a side
transfer for a truthful report. As a result, the principal's burden
of rent provision to the efficient agent is partly transferred to the
inefficient agent. In the optimal contract, side contracting takes place
and the agents exchange side transfers when their types are different.

Second, inducing side transfers between agents of different types
mitigates misreporting incentives when both agents are efficient.
Misreporting by one of the efficient agents, in an attempt to free ride
on the other, is no longer an issue. An efficient agent would never pay
the induced amount of the side transfer to the other agent in order to
free ride because his payoff after paying the side transfer would be
less than the payoff without free riding at all. The other agent,
however, would not let him free ride without receiving the side transfer
(the other agent would also misreport his type without being paid the
side transfer). Thus, the optimal contract under collusion removes the
potential situation in which one agent misreports his type when both
agents are efficient. As a result, the distortion in the output when
reported types are different is recovered. We show that the principal is
better off when the agents are able to collude.

To be sure, this is not the first study to show that collusion
among subunits can be beneficial to an organization. (3) Holmstrom and
Milgrom (1990) and Itoh (1993) argue that collusion between risk-averse agents results in an efficient risk allocation and thus allows the
principal to save on risk compensation. Their studies, unlike ours,
employ moral hazard frameworks in which risk sharing is the source of
the incentive provision to the agents. We employ an adverse selection
model in which information rent is the source of incentive provision. In
our model, collusion between the agents results in an efficient rent
allocation, which enables the principal to improve output efficiency. In
this regard, the current paper is the adverse selection counterpart of
their studies.

In adverse selection models, Tirole (1992), Kofman and Lawarree
(1996), and Lambert-Mogiliansky (1998) consider situations in which
collusion is allowed in the optimal contract. In these studies, however,
the principal allows collusion between the agents because it is too
costly to deter it--the principal's payoff would be higher if the
agents could not collude in the first place. In their models, therefore,
collusion is "detrimental." The studies on
"beneficial" collusion in adverse selection models are those
by Che (1995) and Olsen and Torsvik (1998), who show that the principal
can increase her welfare when the agents can side contract. In Che
(1995), for example, although collusion results in ex post inefficiency,
it is optimal ex ante because under collusion the supervisor has more
incentive to monitor the agent's type in order to receive a bribe.
In their models, if side contracting occurs then hidden information
remains hidden, whereas in our model, side contracting is induced with
the revelation of information in the optimal contract. (4)

This paper is technically related to the models in Martimort
(1997), Laffont and Martimort (1997, 1998), and Baron and Besanko (1992,
1999). The first three papers also study an optimal contract under wage
compression. A key difference between their models and the model in our
paper is that in their models, the outputs from the agents are assumed
to be complements. In such a setting, collusion is detrimental because
wage compression generates negative externality between the different
agents. By contrast, in our model, outputs from the agents are
substitutes and the wage compression generates positive externality.
Using similar models, Baron and Besanko (1992, 1999) compare various
organizational structures. They show that the principal prefers to have
two agents acting like one (consolidation). Unlike ours, however, their
result relies on the assumption that two agents play cooperatively even
before participation--this relaxes the ex post participation constraints for each agent. In our model, the agents can play cooperatively only
after participation (collusion). Collusion is beneficial in our model
not only because wage payments to the agents become more flexible, but
also because the principal can mitigate the misreporting incentives
associated with free-riding opportunities.

Finally, McManus (2001) studies the optimal two-part pricing
strategy to show that a monopolist's profit can increase if the
consumers can share the product through postsale arbitrage among them.
In his paper, however, there is no hidden information, and thus the
monopolist faces no incentive problem. In our paper, the key issue is an
agent's misreporting incentive to the principal in order to free
ride on the other agent--collusion between the agents not only
internalizes the externality, but also mitigates incentive problems.

The rest of the paper is organized as follows. The model is
presented in the next section. In section 3, we discuss the optimal
contract with and without collusion between the agents. We conclude with
some remarks in section 4. All proofs are relegated to the appendices.

2. Model

A risk-neutral principal hires two risk-neutral agents for a
project of a total outcome Q = [q.sup.A] + [q.sup.B], where [q.sup.A]
and [q.sup.B] are the output levels assigned to agents A and B,
respectively. (5) Each agent's type (cost parameter) [beta] can be
low ([[beta].sub.L]) or high ([[beta].sub.H]), with [[beta].sub.L] <
[[beta].sub.H] ([DELTA][beta] [equivalent to] [[beta].sub.H] -
[[beta].sub.L]), Prob([[beta].sub.L]) = [[phi].sub.L],
Prob([[beta].sub.H]) = [[phi].sub.H], and [[phi].sub.L] + [[phi].sub.H]
= 1. The agents' types are stochastically independent and not known
to the principal. The probability distribution of [beta] for each agent
is public information.

The agents do not know each other's type. However, as in
Laffont and Martimort (1997, 1998), we can treat the problem as if they
learn each other's type after participation even though an agent
reports only his own type. That an agent only reports his own type while
he learns the other agent's type is justified in that [beta] is
"soft information," i.e., no verifiable evidence on an
agent's type can be obtained, and hence a court cannot assess it.
(6) Before the agents engage in production, each agent reports his type
[beta] to the principal. The agents can collude in reporting their types
if side contracting is possible. The side contract is assumed to be
enforceable. (7)

After each agent reports his cost parameter [beta], production
takes place. Each agent produces his individual output (q) that
corresponds to his report and sends it to the principal. The output
levels are monitored perfectly, i.e., the principal receives [q.sup.A]
and [q.sup.B] separately. The principal values the total output level Q
= [q.sup.A] + [q.sup.B] according to a strictly concave value function
V(Q), which satisfies the Inada condition. The principal's ex post
payoff is V(Q) - ([t.sup.A] + [t.sup.B]), where [t.sup.A] and [t.sup.B]
are the transfers paid to agents A and B, respectively. The cost of
producing q to an agent is given by [beta]q, and hence each agent's
ex post payoff is t - [beta]q. The agents can quit if they anticipate
that their ex post payoffs are below the reservation level, which is
normalized to zero.

To model wage compression in the simplest way, we assume that the
agents receive equal transfers. (8) Thus, transfer to each agent is tL
when both agents report [[beta].sub.L], [t.sub.M] when one agent reports
[[beta].sub.L] and the other reports [[beta].sub.H], and [t.sub.H] when
both report [[beta].sub.H].

We denote by [q.sub.ij] the individual output level assigned to an
agent reporting [[beta].sub.i] (i = L, H) paired with an agent reporting
[[beta].sub.j] (j = L, H). Therefore, the agents' total output is
denoted by [Q.sub.ij] = [q.sub.ij] + [q.sub.ji] (by symmetry, [Q.sub.ij]
= [Q.sub.ji]). To simplify the notation, we let [Q.sub.H] [equivalent
to] [Q.sub.HH], [Q.sub.M] [equivalent to] [Q.sub.LH], and [Q.sub.L]
[equivalent to] [Q.sub.LL]. We now can express each agent's
individual output in terms of the total output. When the reported types
are the same, both agents produce the same amount: [q.sub.LL] =
[Q.sub.L]/2 and [q.sub.HH] = [Q.sub.H]/2. (9) When the reported types
are different, [q.sub.LH] = r[Q.sub.M] and [q.sub.HL] = (1 - r)[Q.sub.M]
with 1 [greater than or equal to] r [greater than or equal to] 0. That
is, when the agents' types are different, the [[beta].sub.L] agent
produces the proportion r of [Q.sub.M] and the [[beta].sub.H] agent
produces 1 - r of [Q.sub.M]. The contract is contingent on the
agents' reports on their types, and hence it specifies {[Q.sub.L],
[Q.sub.M], r, [Q.sub.H], [t.sub.L], [t.sub.M], [t.sub.H]}. We summarize the timing of the game as follows.

* Each agent learns his type, followed by the contract offer from
the principal.

* The agents accept or refuse the contract.

* Once the contract is accepted, the agents learn each other's
type.

* If side contracting is possible, the agents can collude to
coordinate their reports.

* Reports are made to the principal and the contract is executed
(side transfers are exchanged between the agents if the side contract
took place).

Benchmark. The First-Best Outcome

Before we move on, we look at the first-best outcome. This is the
outcome under perfect wage discrimination and full information. The
first-best outcome is characterized by the following expressions:
V'([Q.sup.*.sub.L]) = [[beta].sub.L], V'([Q.sup.*.sub.M]) =
[[beta].sub.L] with [r.sup.*] = 1, and V'([Q.sup.*.sub.H])=
[[beta].sub.H]. An agent obtains no rent. Notice that when agents of
different types are paired with each other, the [[beta].sub.L] agent
produces the whole [Q.sup.*.sub.M] ([r.sup.*] = 1), and the
[[beta].sub.H] agent gets paid zero transfer. This is an extreme result
due to the constant marginal costs of production, which allows us to
amplify the intuition without altering the main points of the paper.
(10) In the following section, we will discuss the optimal contracts
under hidden information when the principal must pay an equal amount of
transfers to the agents.

3. Results

In this section, we first present the optimal contract when the
agents cannot collude and then derive the optimal contract when
collusion between the agents is possible. We will discuss incentive
issues that cause distortion in the outcome in each case and compare the
optimal outcomes to show that the principal's payoff is higher when
the agents can collude with each other.

Contract under No Collusion ([C.sup.n])

As mentioned before, for expositional purposes, we assume that the
agents learn each other's type after participation, but each agent
reports his own type only ([beta] is soft information, and no verifiable
evidence on an agent's type can be obtained by the other agent).
Since the principal does not know the agents' types, the optimal
contract under no collusion must satisfy the following individual
incentive constraints (hereafter IC):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The ICs above assure an agent that his payoff will be higher when
he reports his type truthfully. For example, ([IC.sup.n.sub.LH])
prevents a [[beta].sub.L] (low cost) agent from exaggerating his cost
parameter when he is paired with a [[beta].sub.H] (high cost) agent.
Since the agents can quit, the optimal contract must satisfy nonnegative payoff of the agents ex post for production to occur. The participation
constraints (hereafter PC) below guarantee the reservation payoff of the
agents in every state:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The optimal contract under no collusion, [C.sup.n], maximizes the
principal's expected payoff,

subject to ([IC.sup.n.sub.LH]) ~ ([PC.sup.n.sub.HH]). The outcome
in [C.sup.n] is characterized in the following lemma.

LEMMA 1. In [C.sup.n] x [Q.sup.n.sub.L] = [Q.sup.*.sub.L],
[Q.sup.n.sub.M] < [Q.sup.*.sub.M], [Q.sup.n.sub.H] <
[Q.sup.*.sub.H]. Moreover, 1/2 < [r.sup.n] < [r.sup.*] (= 1). A
[[beta].sub.L] agent obtains a rent [DELTA][beta](1 - [r.sup.n])
[Q.sup.n.sub.M] if the other agent is [[beta].sub.L], and [DELTA][beta]
[Q.sup.n.sub.H]/2 if the other agent is [[beta.sub.H], while a
[[beta].sub.H] agent obtains no rent. See Appendix A for proof of Lemma
1.

The traditional result known as efficiency at the top (when both
agents are [[beta].sub.L]) holds, and [Q.sub.L] is at its efficient
level. There are, however, downward distortions in [Q.sub.M] and
[Q.sub.H]. To explain the downward distortion in [Q.sub.H], we use
binding ([IC.sup.n.sub.LH]) and ([PC.sup.n.sub.HH]) to express a
[[beta].sub.L] agent's rent when he is paired with a [[beta].sub.H]
agent as

[t.sub.M] - [[beta].sub.L] r[Q.sub.M] = [DELTA][beta][Q.sub.H]/2.

From the equation above, the principal distorts [Q.sub.H] downward
to reduce the rent of a [[beta.sub.L] agent paired with a [[beta].sub.H]
agent. Intuitively, a [[beta].sub.L] agent has an incentive to
exaggerate his cost parameter as [[beta].sub.H] for a higher
compensation. Therefore, by decreasing [Q.sub.H], the principal can
reduce [t.sub.H], which in turn discourages misrepresentation by the
[[beta].sub.L] agent when he is paired with a [[beta].sub.H] agent.

Our main interest in Lemma 1 is the distortion in [Q.sub.M] and r,
the outcome when the different agents receive the same amount of
transfer. To see the distortions, we express the rent of a
[[beta].sub.L] agent (paired with another [[beta].sub.L] agent) as
follows using binding constraints ([IC.sup.n.sub.LL]) and
([PC.sup.n.sub.HL]):

It is clear from Equation 1 that the rent of a [[beta].sub.L] agent
paired with another [[beta].sub.L] agent depends on [Q.sub.M] and r.
From the right-hand side (RHS) of Equation 1, it seems that the
principal can extract the rent by setting r = 1 (without distorting
[Q.sub.M]). This, in fact, would be the case if the principal could
discriminate perfectly the transfers to the agents--when the
agents' types are different, the principal allocates the entire
production to the [[beta].sub.L] agent (r = 1) and pays zero transfer to
the agent reporting [[beta].sub.H]. Under wage compression, however, if
the principal sets r = 1 then she must let the [[beta].sub.H] agent free
ride on the [[beta].sub.L] agent because they both receive [t.sub.M]. In
other words, with r = 1, the [[beta].sub.H]. agent collects [t.sub.M]
for free. Therefore, when both agents are [[beta].sub.L], one of them
may misreport his type in attempt to free ride on the other agent. To
prevent this, the principal removes the potential free-riding
opportunity by setting r < 1. As a result, the RHS of Equation 1
becomes strictly positive, and to reduce the rent to the agents, the
principal distorts [Q.sub.M] downward in the optimal contract.

We summarize the intuition behind the main result without collusion
between the agents. The positive externality between the different
agents resulting from wage compression generates an incentive problem
when both agents are efficient types--an efficient agent has an
incentive to misreport his type to free ride on the other efficient
agent. To discourage an efficient agent from such misrepresentation, the
principal increases (decreases) an inefficient (efficient) agent's
proportion of total production from the efficient level when the types
of the agents are different.

Contract under Collusion ([C.sup.c])

In this subsection, we analyze the structure of the optimal
contract when side contracting between the agents is possible. Although
the possibility of collusion provides the agents with more opportunities
to manipulate their information, it can be advantageous to the principal
because by inducing side transfers, she may be able to make the agents
internalize the positive externality between them.

To explain, we return to the contract under no collusion,
[C.sup.n], as the starting point. Recall that, under no collusion, the
principal distorts r downward (r < 1) to prevent the incentive
problem associated with free-riding opportunities. Suppose the principal
chooses [r.sup.*] and [Q.sup.*.sub.M], the efficient allocation and the
efficient output level when the agents' types are different. Based
on ([IC.sup.n.sub.LH]) in the previous subsection, to induce truth
telling from the [[beta].sub.L] agent, transfer [t.sub.M] must be at
least

where the first term, [[beta].sub.L][r.sup.*][Q.sup.*.sub.M], is
the [[beta].sub.L] agent's production cost and the last term,
[t.sub.H] -- [[beta].sub.L][Q.sub.H]/2, is his information rent. Since
[r.sup.*] = 1, the [[beta].sub.H] agent produces zero while collecting
[t.sub.M]. Now, if the principal reduces [t.sub.M] from the level in
Equation 2, while keeping [r.sup.*] and [Q.sup.*.sub.M], then the
[[beta].sub.L] agent will lose the truth-telling incentive. If the
[[beta].sub.L] agent misreports his type as [[beta].sub.H], however, the
[[beta].sub.H] agent receives no rent from binding (P[C.sup.n.sub.HH]).
Therefore, when side contracting is possible, the [[beta].sub.H] agent
will pay a side transfer to the [[beta].sub.L] agent up to the following
amount for being truthful:

Thus, when the agents are of different types, a burden ([s.sub.M])
of rent provision to the [[beta].sub.L] agent can be transferred from
the principal to the [[beta].sub.H] agent. With the possibility of
inducing side transfers between the agents, the PCs are written as
follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

From the definition of [s.sub.M] in Equation 3, if [s.sub.M] > 0
in (P[C.sup.c.sub.LH]) and ([PC.sup.n.sub.HL]), then there is a positive
side transfer from the [[beta].sub.H] agent to the [[beta].sub.L] agent
and vice versa. If [s.sub.M] = 0 in the optimal contract, then no side
transfer is exchanged between the agents of different types. Notice that
when the agents' types are the same, no side transfer is induced in
the optimal contract. For example, if a side transfer is induced when
both agents are [[beta].sub.H], the PC for one agent would be [t.sub.H]
- [[beta].sub.H][Q.sub.H]/2 + [s.sub.H] [greater than or equal to] 0 and
for the other agent [t.sub.H] - [[beta].sub.H][Q.sub.H]/2 - [s.sub.H]
[greater than or equal to] 0. Clearly, [s.sub.H] = 0 at the optimum.
Similarly, [s.sub.L] = 0 when both agents are [[beta].sub.L].

When the side transfer [s.sub.M] between a [[beta].sub.L] and a
[[beta].sub.H] agent is induced, it affects not only the PCs, but the
ICs as well. The individual incentive constraints that prevent an agent
from misreporting his type are written as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since the side transfer [s.sub.M] may take place in equilibrium,
the ICs become more complex. The side transfer [s.sub.M] does not appear
in the RHS of the ICs because the expressions in the RHSs are an
agent's payoffs from misreporting without side contracting. For
example, in ([IC.sup.c.sub.LL]), a [[beta].sub.L] agent's payoff
from misreporting his type as [[beta].sub.H] is [t.sub.M] -
[[beta].sub.L](1 - r)[Q.sub.M] if the other agent reports [[beta].sub.L]
without being paid [s.sub.M], and [t.sub.H] - [[beta].sub.L][Q.sub.H]/2
if the other agent reports [[beta].sub.H]. Notice that with zero side
transfer ([s.sub.M] = 0), the ICs here become identical to those in
[C.sup.n].

With the possibility of side contracting, the agents would
cooperatively misreport their types if their joint payoff would improve
by doing so. Therefore, the agents have extra room for manipulating
information under collusion, and ICs may fail to induce truth telling
from the agents. To assure truthful reports from the agents, the
principal must impose the following coalition incentive constraints
(hereafter CIC) in her maximization problem:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The RHS of a CIC is the joint payoff of the agents when one or both
of them misreport his/ their type(s) with side contracting, and the CICs
prevent all combinations of misreporting. The positive and negative
flows of the side transfer [s.sub.M] between the agents of different
types cancel each other out in the left-hand side (LHS) of
([CIC.sub.LH,HH]) and ([CIC.sub.LH,LL]), and the RHS of
([CIC.sub.LL,LH]) and ([CIC.sub.HH,LH]).

The optimal contract under collusion, [C.sup.c], maximizes the
principal's expected payoff in (P), subject to ([PC.sup.c.sub.LH])
~ ([CIC.sub.HH,LL]). The outcome in C is characterized below.

The main points in Lemma 2 are that a strictly positive side
transfer occurs with truthful reports and that the efficient allocation
[r.sup.*] and the output level [Q.sup.*.sub.M] are restored. This result
comes from the fact that, under collusion, it is not costly to the
principal to prevent the potential situation in which one of the agents
misreports his type when both agents are [[beta].sub.L]. To see the
intuition behind this result, first consider ([IC.sup.c.sub.LH]), which
is biding in the optimal contract:

From the above equation, a [[beta].sub.L] agent would truthfully
report his type to produce [Q.sub.M] ([??] r = 1) only if the amount of
the side transfer [s.sub.M] (= [t.sub.M] from binding
([PC.sup.c.sub.HL]) with r = 1) is side contracted with the
[[beta].sub.H] agent. This means that the principal transfers the
[[beta].sub.H] agent half the rent provision to the [[beta].sub.L]
agent. Suppose now that both agents are [[beta].sub.L]. If one of the
[[beta].sub.L] agents tries to free ride on the other agent by
misreporting his type as [[beta].sub.H], then he must side contract with
the other [[beta].sub.L] agent to pay the side transfer [s.sub.M] to
him. Otherwise, Equation 4 implies that the other agent will also
misreport his type as [[beta].sub.H]. However, a [[beta].sub.L] agent
has no incentive to pay [s.sub.M] to the other [[beta].sub.L] agent for
free riding (by misreporting his type alone). This is because, as the
following inequality shows, a [[beta].sub.L] agent's payoff after
paying the side transfer [s.sub.M] to the other [[beta].sub.L] agent
becomes lower than his payoff without free riding.

[t.sub.L] - [[beta].sub.L][Q.sub.L]/2 > [t.sub.M] - [s.sub.M](=
0).

Therefore, when side contracting between the agents is possible,
the principal does not need to worry about the potential situation in
which one of the agents misreports his type when both agents are
[[beta].sub.L]. By inducing side contracting between the agents of
different types, the principal can costlessly deter a [[beta].sub.L]
agent from misreporting his type in an attempt to free ride on the other
[[beta].sub.L] agent.

More technically, from binding ([IC.sup.C.sub.LH]) as in Equation
4, the following inequality holds in the optimal contract: [t.sub.M] -
[[beta].sub.L] r[Q.sub.M] < [t.sub.H] - [[beta].sub.L][Q.sub.H]/2.
This inequality implies that binding ([IC.sup.C.sub.LL]) in the optimal
contract is

After substituting for [t.sub.H] by its value (11) in the RHS of
Equations 4 and 5, a [[beta].sub.L] agent's rent is expressed as
[DELTA][beta][Q.sup.c.sub.H]/2 regardless of the other agent's
type. Also, among the CICs, the ones that prevent exaggeration of the
cost parameter(s) are ([CIC.sub.LH,HH]), ([CIC.sub.LL,LH]), and
([CIC.sub.LL,HH]). However, ([CIC.sub.LL,LH]) has no bite in the optimal
contract, which implies that when both agents are [[beta].sub.L], the
principal does not need to worry about a potential situation in which
one of the [[beta].sub.L] agents misreports his type as [[beta].sub.H].
From the RHS of binding ([CIC.sub.LH,HH]) and ([CIC.sub.LL,HH]), the
rent of a [[beta].sub.L] agent is again expressed as
[DELTA][beta][Q.sup.c.sub.H]/2. Since [Q.sub.H] is the only source of
the information rent in every case, although the principal distorts
[Q.sub.H] downward to reduce the rent, she does not need to distort r
and [Q.sub.M] in the optimal contract.

It is also noteworthy that, although [Q.sub.H] is distorted
downward, it is possible that the distortion is smaller than in the case
of no collusion ([Q.sup.c.sub.H] > [Q.sup.n.sub.H]), depending on
parameters. Recall that in [C.sup.n] (under no collusion), [Q.sub.H] is
the source of rent for a [[beta].sub.L] agent only when the other agent
is [[beta].sub.H]. In [C.sup.c] (under collusion), [Q.sub.H] is the
source of rent for a [[beta].sub.L] agent regardless of the type of the
other agent. Therefore, the principal may distort [Q.sub.H] further
downward. However, in [C.sup.c], the principal and a [[beta].sub.H]
agent share rent provision to a [[beta].sub.L] agent, which gives the
principal some room to recover the distortion in [Q.sub.H]. If this
effect is significant enough, then the distortion in [Q.sub.H] under
collusion becomes smaller than the distortion under no collusion.

In summary, when the agents can collude, the principal designs a
contract such that an efficient agent misreports his type unless an
inefficient agent pays a side transfer when they face each other. This
way, the principal can effectively remove free riding by an inefficient
agent, which in turn mitigates an efficient agent's misreporting
incentive to free ride on the other agent.

By comparing the optimal contract in [C.sup.n] and [C.sup.c], we
present our main result in the following proposition.

PROPOSITION 1. The principal's payoff is higher in [C.sup.c]
(under collusion) than her payoff in [C.sup.n] (under no collusion). For
proof of Proposition 1, see Appendix C.

The result above suggests that under wage compression, collusion
between the agents is beneficial to the principal. There are two sources
from which the principal can benefit by inducing side contracting
between agents of different types. First, the principal can make a
[[beta].sub.L] agent and a [[beta].sub.H] agent internalize the
externality between them. Therefore, some burden of rent provision to a
[[beta].sub.L] agent is transferred from the principal to a
[[beta].sub.H] agent. Second, inducing a side transfer between a
[[beta].sub.H] and a [[beta].sub.L] agent automatically prevents a
misreporting incentive when both agents are [[beta].sub.L]. As
mentioned, a [[beta].sub.L] agent would not let the other [[beta].sub.L]
agent misreport his type alone (so that he can free ride) without
receiving the induced amount of the side transfer. However, there will
be no side transfer between the two [[beta].sub.L] agents because one
agent's payoff after paying the side transfer to the other is lower
than his payoff without free riding at all. Therefore, under collusion,
it is no longer the principal's concern that one of the
[[beta].sub.L] agents misreports his type as [[beta].sub.H] to free ride
on the other agent's production. As a result, the distortion in r
and [Q.sub.M] under no collusion can be recovered when collusion is
possible.

4. Conclusion with Remarks

In this paper, we have presented a model in which side contracting
between agents improves the principal's welfare. In our model,
hidden information and the free-riding problem interact with each other
because wage discrimination against the agents of different types is
limited. We have shown that, under collusion, the principal can reduce
the amount of the rent provision by inducing an inefficient agent to
bribe an efficient agent, who in turn removes a misreporting incentive
for free riding when both agents are efficient. As a result, the output
distortions associated with hidden information are partly recovered, and
the principal's payoff becomes higher under collusion between the
agents.

We close this paper with several remarks. First, if the wage
transfers to the different agents can be perfectly discriminated, then,
as usual, the principal's payoff is higher under no collusion. In
our model, limited wage discrimination does not affect the optimal
outcome if the principal only needs to satisfy the ex ante participation
constraints for the agents (i.e., the agents cannot quit). In such a
setting, the principal can let an inefficient agent free ride when the
other agent is efficient, but makes his ex post rent negative when the
other agent is also inefficient (thus the expected rent of an
inefficient agent is zero). Similarly, the principal lets an efficient
agent enjoy a relatively large rent when the other agent is also
efficient, but pays a relatively small transfer when the other agent is
inefficient. Since all parties are risk neutral, wage compression with
such averaging out has the same effect ex ante as perfect wage
discrimination, and the principal prefers the outcome under no
collusion. Second, for expositional purpose, we assumed that there is no
transaction cost related to side transfers between the agents. In our
model, the principal always will induce side contracting unless the
transaction cost is less than the amount of the side transfer. (12)
Third, as mentioned before (footnote 4), hiring multiple agents in our
model is justified because the principal can hire an efficient agent
with a higher probability--it is possible that the principal wants to
hire only one agent depending on the parameters. (13) Finally, although
the possibility of collusion is beneficial to the principal, it may not
be for the agents. (14) In such cases, if the agents have commitment
power before the principal's contract offer, they may commit to
avoid collusion after participation.

Appendix A

PROOFOF LEMMA 1. Since the objective function is concave and the
constraints are linear thus convex, the solution is unique. Thus, we
show that ([PC.sup.n.sub.HH]), ([PC.sup.n.sub.HL]), ([IC.sup.n.sub.LH]),
and ([IC.sup.n.sub.LL]) are binding in deriving the solution. It is
straightforward to show that the other constraints are satisfied with
the solution without them. With ([PC.sup.n.sub.HH]),
([PC.sup.n.sub.HL]), ([IC.sup.n.sub.LH]), and ([IC.sup.n.sub.LL]), the
Lagrangian is written as

Likewise, the transfers are obtained by the binding constraints
([PC.sup.n.sub.HH]), ([IC.sup.n.sub.LH]), and ([IC.sup.n.sub.LL]).
Substituting for r and the transfers by their values in the objective
function and differentiating with respect to [Q.sub.L], [Q.sub.M], and
OH yields

From the above expressions, [Q.sup.n.sub.L] = [Q.sup.*.sub.L],
[Q.sup.n.sub.M] < [Q.sup.*.sub.M], [Q.sup.n.sub.H] <
[Q.sup.*.sub.H]. Since [Q.sup.n.sub.L] > [Q.sup.n.sub.M] >
[Q.sup.n.sub.H], together with the expression in Equation A12, we can
verify that 1/2 < [r.sup.n] < 1 (= [r.sup.*]). The rent of each
agent is obtained from binding constraints ([PC.sup.n.sub.HH]),
([IC.sub.n.sub.LH]), and ([IC.sup.n.sub.LL]). QED.

Appendix B

PROOF OF LEMMA 2. AS usual in the model of this type, the CICs
encompass the ICs since the CICs prevent all combinations of
misreporting. Thus, we construct the Lagrangian with ([CIC.sub.LH,HH]),
([CIC.sub.LL,HH]), and ([PC.sub.HH]) and show that these constraints are
binding in deriving the optimal outcome. It is straightforward to verify
that other constraints are satisfied by our solution. The Lagrangian is

From Equation B4, [theta] = [[phi].sup.2.sub.L] (>0), and hence
Equation B8 implies that ([CIC.sub.LL,HH]) is binding. From Equation B5,
we have [sigma] = 2[[phi].sub.L][[phi].sub.H], and binding
([CIC.sub.LH,HH]) is implied by Equation B7. Also from Equation B6,
[omega] = 2[[phi].sup.2.sub.H] + 2[sigma] + 2[theta] (>0). Therefore,
Equation B9 implies that ([PC.sup.c.sub.HH]) is binding. Differentiating
the Lagrangian with respect to r gives [sigma]([[beta].sub.H]
[[beta].sub.L])[Q.sub.M] 0, which implies that r = 1 at the optimum. The
transfers and each agent's rent are obtained from the binding
constraints ([CIC.sub.LH,HH]), ([CIC.sub.LL,HH]), and ([PC.sub.HH]), and
with r = 1 in Equation 3, we have [S.sub.M] = [t.sub.M] > 0.
Replacing the transfers with their values in the objective function and
differentiating with respect to [Q.sub.L], [Q.sub.M], and [Q.sub.H]
yields

By comparing [[PHI].sup.c.sub.A] and [[PHI].sup.n], it is clear
that [[PHI].sup.c.sub.A] > [[PHI].sup.n], which is followed by
[[PHI].sup.c] > [[PHI].sup.n]. QED.

I would like to thank Ingela Alger, Fahad Khalil, Jacques Lawarree,
two anonymous referees, and the editor for detailed comments and
suggestions. I also thank Ching-To Albert Ma, Bill Sundstrom, Gerald Roland, and the seminar participants at the 2006 spring Midwest Economic
Theory Conference for helpful comments. Received March 2005; accepted
October 2006.

Eccles, Robert. 1985. Transfer pricing as a problem of agency. In
Principals and agents: The structure of business, edited by John Prett
and Richard Zeckhauser. Cambridge, MA: Harvard University Press.

(1) See Baron and Kreps (1999). Eccles (1985), in his empirical
findings, shows that organizations often face a "fairness"
restriction because subunits in similar positions within an
organizational hierarchy see as unfair the fact that they may receive
different transfers resulting from exogenous parameters.

(2) As Mintzberg (1983) notes, not only direct monetary exchanges,
but also sharing resources with other subunits, can be interpreted as
side transfers.

(3) As mentioned above, collusion among agents typically limits the
welfare of the principal. See Tirole (1986), Kofman and Lawarree (1993),
and Kessler (2000) for examples.

(4) See also Lawarree and Shin (2005) for information revelation
when side contracting takes place. In their study, however, side
contracting is not actively induced by the principal as in the current
paper.

(5) Since outputs are perfect substitutes, the principal may want
to hire one agent. Hiring multiple agents is justified by increasing the
likelihood of having an efficient agent.

(6) Therefore, we are implicitly assuming that if an agent's
type is also reported by the other agent, an agent has a right to
protest the other agent's report in the court of law, and
resolution is prohibitively costly, for there is no hard evidence.

(7) This is a standard assumption. See Tirole (1992) for a
discussion of enforceability of side contracts.

(8) Our main result holds as long as the principal cannot perfectly
discriminate the transfers.

(9) In the principal's problem, constraints are identical for
agents of the same type. which implies that [q.sub.LL] = [Q.sub.L]/2 and
[q.sub.HH] = [Q.sub.H]/2.

(10) For example, our main result holds with quadratic cost
functions. As mentioned before, hiring multiple agents is justified by
increasing the likelihood of having an efficient agent.

(14) Since a [[beta].sub.H] agent's rent is zero in both
[C.sub.n] and [C.sub.n], the agents are weakly better off in [C.sup.c]
when the following inequality holds for a [[beta].sub.L] agent's
rent: [[phi].sub.L](1 - [r.sup.n]) [Q.sup.n.sub.M] +
[[phi].sub.H][Q.sup.n.sub.H]/2 < [Q.sup.c.sub.H]/2.